--- a/thys3/RegLangs.thy Wed Feb 15 11:52:22 2023 +0000
+++ b/thys3/RegLangs.thy Thu Feb 16 23:23:22 2023 +0000
@@ -1,4 +1,3 @@
-
theory RegLangs
imports Main "HOL-Library.Sublist"
begin
@@ -22,6 +21,42 @@
and "{} ;; A = {}"
by (simp_all add: Sequ_def)
+lemma concI[simp,intro]: "u : A \<Longrightarrow> v : B \<Longrightarrow> u@v : A ;; B"
+by (auto simp add: Sequ_def)
+
+lemma concE[elim]:
+assumes "w \<in> A ;; B"
+obtains u v where "u \<in> A" "v \<in> B" "w = u@v"
+using assms by (auto simp: Sequ_def)
+
+lemma concI_if_Nil2: "[] \<in> B \<Longrightarrow> xs : A \<Longrightarrow> xs \<in> A ;; B"
+by (metis append_Nil2 concI)
+
+lemma conc_assoc: "(A ;; B) ;; C = A ;; (B ;; C)"
+by (auto elim!: concE) (simp only: append_assoc[symmetric] concI)
+
+
+text \<open>Language power operations\<close>
+
+overloading lang_pow == "compow :: nat \<Rightarrow> string set \<Rightarrow> string set"
+begin
+ primrec lang_pow :: "nat \<Rightarrow> string set \<Rightarrow> string set" where
+ "lang_pow 0 A = {[]}" |
+ "lang_pow (Suc n) A = A ;; (lang_pow n A)"
+end
+
+
+lemma conc_pow_comm:
+ shows "A ;; (A ^^ n) = (A ^^ n) ;; A"
+by (induct n) (simp_all add: conc_assoc[symmetric])
+
+lemma lang_pow_add: "A ^^ (n + m) = (A ^^ n) ;; (A ^^ m)"
+ by (induct n) (auto simp: conc_assoc)
+
+lemma lang_empty:
+ fixes A::"string set"
+ shows "A ^^ 0 = {[]}"
+ by simp
section \<open>Semantic Derivative (Left Quotient) of Languages\<close>
@@ -89,6 +124,11 @@
unfolding Der_def Sequ_def
by(auto simp add: Star_decomp)
+lemma Der_inter[simp]: "Der a (A \<inter> B) = Der a A \<inter> Der a B"
+ and Der_compl[simp]: "Der a (-A) = - Der a A"
+ and Der_Union[simp]: "Der a (Union M) = Union(Der a ` M)"
+ and Der_UN[simp]: "Der a (UN x:I. S x) = (UN x:I. Der a (S x))"
+by (auto simp: Der_def)
lemma Der_star[simp]:
shows "Der c (A\<star>) = (Der c A) ;; A\<star>"
@@ -104,6 +144,13 @@
finally show "Der c (A\<star>) = (Der c A) ;; A\<star>" .
qed
+lemma Der_pow[simp]:
+ shows "Der c (A ^^ n) = (if n = 0 then {} else (Der c A) ;; (A ^^ (n - 1)))"
+ apply(induct n arbitrary: A)
+ apply(auto simp add: Cons_eq_append_conv)
+ by (metis Suc_pred concI_if_Nil2 conc_assoc conc_pow_comm lang_pow.simps(2))
+
+
lemma Star_concat:
assumes "\<forall>s \<in> set ss. s \<in> A"
shows "concat ss \<in> A\<star>"
@@ -129,6 +176,7 @@
| SEQ rexp rexp
| ALT rexp rexp
| STAR rexp
+| NTIMES rexp nat
section \<open>Semantics of Regular Expressions\<close>
@@ -141,7 +189,7 @@
| "L (SEQ r1 r2) = (L r1) ;; (L r2)"
| "L (ALT r1 r2) = (L r1) \<union> (L r2)"
| "L (STAR r) = (L r)\<star>"
-
+| "L (NTIMES r n) = (L r) ^^ n"
section \<open>Nullable, Derivatives\<close>
@@ -154,7 +202,7 @@
| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)"
| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)"
| "nullable (STAR r) = True"
-
+| "nullable (NTIMES r n) = (if n = 0 then True else nullable r)"
fun
der :: "char \<Rightarrow> rexp \<Rightarrow> rexp"
@@ -168,6 +216,8 @@
then ALT (SEQ (der c r1) r2) (der c r2)
else SEQ (der c r1) r2)"
| "der c (STAR r) = SEQ (der c r) (STAR r)"
+| "der c (NTIMES r n) = (if n = 0 then ZERO else SEQ (der c r) (NTIMES r (n - 1)))"
+
fun
ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp"
@@ -176,13 +226,23 @@
| "ders (c # s) r = ders s (der c r)"
+lemma pow_empty_iff:
+ shows "[] \<in> (L r) ^^ n \<longleftrightarrow> (if n = 0 then True else [] \<in> (L r))"
+ by (induct n) (auto simp add: Sequ_def)
+
lemma nullable_correctness:
shows "nullable r \<longleftrightarrow> [] \<in> (L r)"
-by (induct r) (auto simp add: Sequ_def)
+ by (induct r) (auto simp add: Sequ_def pow_empty_iff)
lemma der_correctness:
shows "L (der c r) = Der c (L r)"
-by (induct r) (simp_all add: nullable_correctness)
+ apply (induct r)
+ apply(auto simp add: nullable_correctness Sequ_def)
+ using Der_def apply force
+ using Der_def apply auto[1]
+ apply (smt (verit, ccfv_SIG) Der_def append_eq_Cons_conv mem_Collect_eq)
+ using Der_def apply force
+ using Der_Sequ Sequ_def by auto
lemma ders_correctness:
shows "L (ders s r) = Ders s (L r)"
@@ -198,40 +258,4 @@
by (simp add: ders_append)
-(*
-datatype ctxt =
- SeqC rexp bool
- | AltCL rexp
- | AltCH rexp
- | StarC rexp
-
-function
- down :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"
-and up :: "char \<Rightarrow> rexp \<Rightarrow> ctxt list \<Rightarrow> rexp * ctxt list"
-where
- "down c (SEQ r1 r2) ctxts =
- (if (nullable r1) then down c r1 (SeqC r2 True # ctxts)
- else down c r1 (SeqC r2 False # ctxts))"
-| "down c (CH d) ctxts =
- (if c = d then up c ONE ctxts else up c ZERO ctxts)"
-| "down c ONE ctxts = up c ZERO ctxts"
-| "down c ZERO ctxts = up c ZERO ctxts"
-| "down c (ALT r1 r2) ctxts = down c r1 (AltCH r2 # ctxts)"
-| "down c (STAR r1) ctxts = down c r1 (StarC r1 # ctxts)"
-| "up c r [] = (r, [])"
-| "up c r (SeqC r2 False # ctxts) = up c (SEQ r r2) ctxts"
-| "up c r (SeqC r2 True # ctxts) = down c r2 (AltCL (SEQ r r2) # ctxts)"
-| "up c r (AltCL r1 # ctxts) = up c (ALT r1 r) ctxts"
-| "up c r (AltCH r2 # ctxts) = down c r2 (AltCL r # ctxts)"
-| "up c r (StarC r1 # ctxts) = up c (SEQ r (STAR r1)) ctxts"
- apply(pat_completeness)
- apply(auto)
- done
-
-termination
- sorry
-
-*)
-
-
end
\ No newline at end of file